Poincaré Plots in Dynamical Systems

• Poincaré plots are a method that reduces high-dimensional flows by recording trajectory intersections with a transversal hypersurface to analyze orbit types.
• They employ numerical integration, sparse regression, and image-based techniques to distinguish regular, chaotic, and ballistic regimes with practical precision.
• Applications across astrodynamics, plasma physics, and biomedical signals offer actionable insights into invariant manifolds, stability, and transport phenomena.

A Poincaré plot, often termed a Poincaré section or Poincaré map, is a central tool in the analysis of deterministic dynamical systems. It consists of the intersection set between a continuous or discrete-time trajectory and a lower-dimensional, transversal hypersurface in phase space, effectively reducing the paper of high-dimensional flows to discrete iterated dynamics or maps. This tool is fundamental in distinguishing regular, quasi-periodic, and chaotic regions in both theoretical and applied contexts, and underpins a range of computational and data-driven techniques for analyzing the qualitative structure and transport properties of nonlinear systems.

1. Mathematical Definition and Construction of Poincaré Plots

A Poincaré plot results from the systematic strobing or sectioning of a flow by a co-dimension-one hypersurface, typically defined as

$$S = \{ x \in \mathbb{R}^{d+1}: h(x) = 0 \}, \quad \nabla h \neq 0,$$

chosen transverse to the vector field, with the property that $\nabla h(x)\cdot f(x)\neq 0$ on $S$ (Bramburger et al., 2019, Crossley et al., 2021). For continuous-time systems $\dot x = f(x)$, each time a trajectory $x(t)$ crosses $S$ in the prescribed direction, the intersection is recorded: $x_n := x(t_n) \in S.$ The map

$x_{n+1} = F(x_n),$

defines the Poincaré map on the $d$-dimensional section $S$.

For discrete-time systems (e.g., symplectic maps), every iteration is an intersection, and the Poincaré plot comprises the sequential points $(p_n, x_n)$ as generated by the underlying map (Bósio et al., 19 Sep 2024, Ruth et al., 2023). For Hamiltonian flows, the section is often chosen to exploit symmetries or periodicities (e.g., $y \bmod 2\pi=0$ with fixed crossing direction), ensuring that regular (KAM) orbits form invariant curves or islands in $(p,x)$, while chaotic orbits fill regions densely (Bósio et al., 19 Sep 2024).

The choice of section is not unique and can strongly influence the observed topology of the map. In high-dimensional reconstructed attractors (e.g., via Takens' embedding from biophysical signals), the section is typically a coordinate plane at constant value (Mukherjee et al., 2014). The process includes interpolation to recover the precise intersection point between sampled trajectory points.

2. Computational Methodologies for Poincaré Plot Generation

Several algorithmic approaches have been developed for constructing Poincaré plots:

• Direct Numerical Integration: As in the classical 4th-order Runge–Kutta schemes, with zero-crossing detection and (possibly linear) interpolation used to locate section intersections in continuous flows (Crossley et al., 2021). Trajectories are integrated from grids of initial conditions, and ensembles of intersection points reveal the phase space structure. In standard and time-dependent Hamiltonian models, this involves populating grids of $M$ initial conditions, propagating each for $O(10^4)$ steps, and rasterizing intersecting points into high-resolution images (Bósio et al., 19 Sep 2024).
• Sparse Regression and Data-Driven Inference (SINDy): When only time-series data is available, library-based sparse regression (SINDy) is used to construct the map

$x_{k+1} \approx \Xi^T \Theta(x_k),$

with $\Theta$ a matrix of candidate nonlinear functions (usually polynomials, trigonometric terms), and $\Xi$ a sparse coefficient matrix. LASSO or sequential thresholded least squares are typical for enforcing sparsity (Bramburger et al., 2019).

• Image-Based Techniques: Phase-space points are binned into pixels at fixed resolution. Morphological image filtering distinguishes holes (chaotic regions), thin structures (invariant curves), and connected components (ballistic islands). Cores of each connected region are extracted, and representative orbits from these points are used to classify the transport regime via additional short trajectory integrations (Bósio et al., 19 Sep 2024).
• Kernel-Based Invariant Functions: For symplectic maps, an invariant scalar function $f$ is learned via radial basis kernel regression, such that $f(\varphi(x)) \approx f(x)$. The level sets of $f$ reconstruct nearly invariant tori and islands, and the pointwise residual $r(x)=f(\varphi(x))-f(x)$ diagnoses local chaos or map complexity (Ruth et al., 2023).

These diverse methodologies allow for both direct and indirect construction of Poincaré plots, subject to computational trade-offs, sample efficiency, and required accuracy.

3. Extraction, Classification, and Analysis of Phase-Space Structures

Poincaré plots reveal core dynamical structures, and their effective characterization is essential for understanding transport and recurrence.

• Classification of Orbit Types: Automated image processing allows for rapid classification into ballistic (accelerator modes; displacement $\langle(\Delta x)^2\rangle\sim n^2$), regular (small standard deviation in rotation number), unbounded chaotic (large displacement but non-ballistic), and bounded chaotic (non-regular, non-ballistic, confined) regimes (Bósio et al., 19 Sep 2024).
• Invariant Manifold Extraction: In applications such as the planar CR3BP, adaptive subdivision and winding-number analysis are used to detect fixed points (periodic orbits), their stability, and associated stable/unstable manifolds (the "topological skeleton") (Tricoche et al., 2020). Numerical eigen-analysis of the monodromy matrix or Jacobian of the return map classifies local stability:

$$|\lambda|<1 \text{ (stable)}, \quad |\lambda|>1 \text{ (unstable)}, \quad |\lambda|=1 \text{ (center)}.$$

• Residual-Based Chaos Diagnostics: Kernel-based labeling functions $f$ provide a smooth residual field $r(x)$, directly visualizing the onset and extent of chaos and mapping invariant structures (Ruth et al., 2023).
• Lyapunov Exponents of the Map: For reconstructed data-driven maps, computation of Lyapunov exponents from Jacobian products quantifies chaos (one exponent $>0$, one $<0$ for 2D maps) and verifies fidelity of the discrete map to true continuous dynamics (Mukherjee et al., 2014).

4. Rapid Detection of Transport Phenomena and Regime Boundaries

Poincaré plots support fast detection and mapping of important dynamical transitions:

• Ballistic and Superdiffusive Modes: In the standard map, the presence of ballistic (accelerator) modes is indicated by distinct islands in the Poincaré image; visual methods detect these regimes an order of magnitude faster than mean square displacement-based ensemble calculations. A nonzero area of ballistic regions in the image correlates directly with superdiffusion and is detected long before ensemble averages converge (Bósio et al., 19 Sep 2024).
• Parameter Scans and Regime Transitions: The image-based and kernel-based approaches facilitate efficient scanning across high-dimensional parameter spaces, mapping regions of chaos, regularity, and ballistic transport. In stellarator confinement or multi-wave Hamiltonian systems, the approach enables practical device optimization and experimental comparison (Ruth et al., 2023, Bósio et al., 19 Sep 2024).
• Mapping Onset of Global Chaos: The transition in minimized residuals of kernel-invariant functions or sudden changes in the structure of Poincaré plots (e.g., at $k_c \approx 0.97$ for the standard map) succinctly identify the breakup of invariant tori and the onset of global chaos (Ruth et al., 2023).

5. Applications in Physics, Astrodynamics, and Signal Analysis

Poincaré plots are broadly used across fields:

• Astrodynamics: In trajectory design for the circular restricted three-body problem, extraction of fixed points, invariant manifolds, and their intersections from Poincaré plots allows for efficient construction of low-energy transfer orbits, uncovering previously unknown periodic orbits and providing interactive design frameworks (Tricoche et al., 2020). The topological structures inform both qualitative understanding and practical mission planning.
• Hamiltonian and Plasma Systems: Visualization and analysis of magnetic islands, chaos, and transport barriers in plasma confinement are routinely performed via Poincaré sections of symplectic maps (Ruth et al., 2023).
• Time-Series and Biomedical Signals: Embedded attractors from scalar time series (e.g., EMG data) can be sectioned with Poincaré maps to reveal underlying low-dimensional (possibly chaotic) dynamics; the choice of section critically determines whether chaos is detected in the map (Mukherjee et al., 2014).

6. Limitations, Extensions, and Alternative Techniques

Poincaré plots, while powerful, have inherent limitations:

• Dimensionality: Visualization and effective map construction degrade for systems with more than two degrees of freedom (2D section), as higher-dimensional maps become hard to interpret (Crossley et al., 2021).
• Resolution and Filtering: Image-based approaches may miss small-scale structures due to finite pixel size and filter parameters (Bósio et al., 19 Sep 2024). Morphological operations require system-dependent tuning.
• Sample Complexity: Long integration times or high sampling density are required in traditional approaches; kernel methods and image-based strategies mitigate this at increased computational or memory cost (Ruth et al., 2023, Bósio et al., 19 Sep 2024).
• Section Dependency: The choice of section can hide or reveal structures and chaos. Only certain cuts will faithfully reflect the exponential divergence or invariant sets of the flow (Mukherjee et al., 2014).
• Comparisons with Lagrangian Descriptors: Alternative approaches, such as Lagrangian Descriptors (LDs), provide scalar fields over sections that reveal invariant manifolds without explicit map construction. LDs may handle higher-dimensional, time-dependent, or stochastic flows more naturally, but interpretation can be less direct for some users (Crossley et al., 2021).

Potential extensions include adaptive or multiscale filtering, local Lyapunov diagnostics, high-dimensional kernel approximations, and application to experimental data and higher-dimensional systems.

7. Summary and Outlook

The contemporary repertoire of techniques for constructing, analyzing, and interpreting Poincaré plots leverages advances in kernel-based regression, image processing, sparse modeling, and adaptive numerical integration. These methods enhance the classical role of Poincaré sections in revealing the skeleton of nonlinear dynamics, supporting the classification of transport regimes, and enabling efficient parameter-space exploration across application domains (Bósio et al., 19 Sep 2024, Ruth et al., 2023, Tricoche et al., 2020, Mukherjee et al., 2014, Bramburger et al., 2019).

Ongoing research aims to scale these approaches to higher-dimensional systems, improve accuracy in small-scale structure detection, and further integrate data-driven methodologies with domain physics. The Poincaré plot remains a foundational construct in nonlinear dynamics, chaos, and transport, continuously informing both theoretical advancements and practical design of complex systems.